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2 68 Asian Journal of Control, Vol. 6, No. 2, pp , March 204 and g (x ) are nown nonlinear functions of x and. he process noise h R n and measurement noise v R p are assumed to be uncorrelated and normally distributed with covariance Q and R respectively. CQKF approximately evaluates the intractable integrals encountered in Baysian filtering framewor using cubaturequadrature points and corresponding weights associated with them. For n dimensional problem solved with third order spherical cubature and n order of quadrature rule, total number of points and associated weights need to be calculated. he algorithm of the cubature-quadrature Kalman filter (CQKF) could be summarized as follows: Step i. Filter initialization. Initialize the filter with xˆ 00 and P 0 0. Calculate the cubature-quadrature (CQ) points x j and their corresponding weights w j( j =, 2,..., ). Step ii. Predictor step. Perform Cholesy decomposition of posterior error covariance P = S S (3) = S ξ + xˆ (4) j, j Update cubature-quadrature points φ( ) (5) j, + = j, Compute time updated mean and covariance ˆx = w (6) + j j, + + j[ j, + + ] j, + + [ ] + P = w xˆ xˆ Q (7) Step iii. Corrector step or measurement update. Perform Cholesy decomposition of prior error covariance P = S S (8) = S ξ + xˆ (9) j, + + j + where j =,2,.... Find the predicted measurements at each cubaturequadrature points Y j, + = j, + γ( ) (0) Estimate the predicted measurement ŷ = w Y, () + j j + Calculate the covariances [ ] + Py y = wj Yj y Yj y R + +, + ˆ +, + ˆ + (2) [ ] Px y = wj j x Yj y + +, + ˆ +, + ˆ + (3) Calculate Kalman gain K P P [ ][ ] + = x+ y+ y+ y (4) + Compute posterior state values xˆ+ + = xˆ+ + K+ ( y+ yˆ+ ) (5) Posterior error covariance matrix is given by P+ + = P+ K+ Py y K (6) III. SQUARE-ROO CUBAURE-QUADRAURE KALMAN FILER he SR-CQKF is an algebraic re-formulation of CQKF algorithm, where Cholesy decomposition need not to be performed at each step. Due to limited word length in processing software, round off error accumulates and after some iterations the positive semi-definite nature of the error covariance matrix may be lost. o circumvent the problem, in this section, we re-formulate the CQKF algorithm based on orthogonal-triangular decomposition, popularly nown as QR decomposition. he error covariance matrix, P can be factorized as, P = AA (7) where P R n n, and A R n m with m n. Due to increased dimension of A, Arasaratnam and Hayin [7] called it as fat matrix. he QR decomposition of matrix A is given by, A = QR where Q R m m is orthogonal, and R R m n is upper triangular. With the QR decomposition described above, the error covariance matrix can be written as P = AA = R Q QR= R R = SS (8) R is the upper triangular part of R matrix, n n R R, where R = S. he detailed algorithm of square-root cubature-quadrature Kalman filter is provided below. We use the notation qr{} to denote the QR decomposition of a matrix and uptri{} to select the upper triangular part of a matrix. 203 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

3 S. Bhaumi and Swati: Square-Root Cubature-Quadrature Kalman Filter 69 Step i. Filter initialization. Initialize the filter with xˆ 00 and P00 = S00 S00 Calculate the cubature-quadrature (CQ) points x j and their corresponding weights w j (j =, 2,..., ). he methodology to calculate n order cubature quadrature points and associated weights has been described in paper [3]. Step ii. Predictor step. = S ξ + xˆ (9) j, j Update cubature-quadrature points φ( ) (20) j, + = j, Compute time updated mean ˆx = w (2) + j j, + Evaluate square-root weight matrix W = w 0 0 w2 nn (22) Calculate the matrix of square-root weighted cubature-quadrature points after prior mean subtraction [ ] * = xˆ xˆ xˆ W +, + + 2, + +, + + Perform QR decomposition { } (23) R = + qr * + Q (24) Estimate the square-root of predicted error covariance S = uptri{ R } (25) + + Step iii. Measurement update. = S ξ + xˆ (26) j, + + j + Find the predicted measurements at each CQ points Y j, + = j, + γ( ) (27) Estimate the predicted measurement ŷ = w Y, (28) + j j + Calculate the square-root weighted measurement matrix [ ] Y* = Y yˆ Y yˆ Y yˆ W +, + + 2, + +, + + Perform QR decomposition { } (29) R = y + y qr Y * + + R (30) Calculate the square root of innovation covariance S = uptri{ R } (3) y+ y+ y+ y+ Calculate the cross covariance matrix P * Y* (32) x y = Calculate Kalman gain K P S S = ( ) x y y y y y (33) Compute posterior state values xˆ+ + = xˆ+ + K( y+ yˆ+ ) (34) Calculate QR decomposition { } R = + + qr * + KY * + Q K R (35) Calculate square root of posterior error covariance matrix S = uptri{ R } (36) Note. he equation (35) can be derived from equations (4), (6), (24), (30), and (32) by matrix manipulation. For detailed steps readers are refered to [7]. Note 2. he practitioners who want to implement the algorithm in MALAB software environment, should note that the command chol (P) returns the matrix S (where P = SS ) rather than S. Accordingly the expressions for next steps need to be evaluated. IV. SIMULAION RESULS In this section, the formulated algorithm described above has been applied to solve a single dimensional as well as multi dimensional nonlinear problem. Example. Single dimensional process. he plant model, inspired by [0], has one unstable and two stable equilibrium points and given by x+ = φ( x) + η 2 2 φ( x) = x+ Δt5x( x ), η ~ ℵ( 0, b Δt) 203 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

4 620 Asian Journal of Control, Vol. 6, No. 2, pp , March 204 he measurement model is y = γ ( x ) + v 2 γ ( x) = Δtx( 0. 5x), v ~ ℵ( 0, d Δt) and the value of Dt = 0.0 seconds, x 0 =-0.2, xˆ 00 = 08., P 0 0 = 2, b = 0.5 and d = 0.. We have considered the time span from 0 to 4 seconds. he system has three equilibrium points, of which, the one at the origin is unstable and the other two are at and stable. In the absence of any bias, the system settles around either of the two stable equilibrium points. he problem becomes challenging because moderate estimation error forces the estimate to settle down at the wrong equilibrium point, leading to a trac loss situation. We solve the estimation problem using and SR-CQKF. he percentage fail counts for different filters are shown in able I. he percentage fail count is defined as the number of cases where estimation error at 4th second is more than unity out of 00 Monte Carlo runs. In our notation, CQKF-x indicates an x order Gauss-Laguerre cubature-quadrature filter. Under the first order Gauss-Laguerre approximation CQKF reduces to CKF. he numbers mentioned in the table show the improvement of performance with the square-root CQKF. able I. Percentage fail count. Filter Fail Count 22% or SR-CQKF- 4%.05%.02%.02% Fig. shows the root mean square error () obtained from SR-CQKF and out of 0,000 Monte Carlo runs. It can be seen from Fig., that of higher order SR-CQKF converges more quicly than first order SR-CQKF or. However no significant improvement of performance is noticed with more than second order SR-CQKF. Example 2. Lorenz system. Inspired from earlier wors [0,], in this example, we consider the discrete time Lorenz attractor, one of the classic icons in nonlinear dynamics. he attractor is available in the literature from 963, mainly due to the wor of meteorologist Edward Lorenz. Although a higher dimensional Lorenz attractor may exists [2,3] in real life problems, here we consider three dimensional system. he process equation is given by x+ = φ( x) + bη φ( x) = x+ Δtf( x), η ~ ℵ( 0, Δt) he measurement equation is expressed as y = γ ( x ) + dv γ ( x) = Δth( x), v ~ ℵ( 0, Δt) Here x = [x, x 2, x 3,] R 3 is state variable. he parameters b R 3, and d R are constant whose values are taen as b = [ ] and d = he functions f(x) and h(x) are given by [ ] f( x) = α( x + x ) βx x x x γx + x x ime (s) Fig.. plot of SR-CQKF with higher order Gauss-Laguerre quadrature points. 203 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

5 S. Bhaumi and Swati: Square-Root Cubature-Quadrature Kalman Filter 62 able II. Average root mean square error over time span. Filter Averaged Averaged Averaged (State ) (State 2) (State 3) or SR-CQKF and ime (sec) Fig. 2. plot of and SR-CQKF for first state variable. h( x)= x 2 + x2 2 + x3 2 he three parameters, a (called the Prandtl number), b (called the Rayleigh number), and g have great impact on the system. he system has three unstable equilibrium points and for a 0, and g(b - ) 0 they are at [0 0 0], γ( β ) γ( β ) ( β ), and γ( β ) γ( β ) ( β ). We chose the system with classical parameter values: a = 0, g = 8/3, and b = 28, for which almost all points in phase space tend to a strange attractor [3]. he initial truth value of state is taen as x 0 = [ ]. We simulate during the time interval of 0 to 4 seconds with the sampling time Dt = 0.0 seconds. he initial posterior estimate is assumed as xˆ 00 = [ ] with initial error covariance P 0 0 = 0.35I 3. he above described problem has been solved using square-root version of CQKF with higher order of Gauss-Laguerre approximation. We observe that sometimes ordinary CQKF fails to run due to non positive definite error covariance matrix. he same problem does not occur during SR-CQKF as expected from the formulation. he root mean square error, averaged over time span, obtained from 50 Monte Carlo runs, has been tabulated in able II. In Figs 2 4, we show the root mean square errors (s) of the states obtained from and SR-CQKF. was calculated from 50 Monte Carlo ime (sec) Fig. 3. plot of and SR-CQKF for second state variable ime (sec) Fig. 4. plot of and SR-CQKF for third state variable. runs. he results obtained from the simulation run may be summarized as follows: From Fig. 4, no improvement in estimation of third state variable has been observed with SR-CQKF in comparison with. his is also supported from the numbers displayed in able II. uses first order linearization to approximately calculate the mean and covariance of non Gaussian probability density function. Cubature quadrature Kalman filter uses cubature rule and Gauss-Laguerre quadrature points to 203 Chinese Automatic Control Society and Wiley Publishing Asia Pty Ltd

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